**C(q) **is the least amount of money needed to buy inputs
that will produce output q. It is the cost function.

**FC **are fixed costs, the costs incurred even if there is
no production. FC = C(0).

**VC(q)** are variable costs. VC(q) = C(q) - FC.

**AC(q)** is average cost. AC(q) = C(q)/q.

**AVC** is average variable cost. AC(q) = VC(q)/q.

**AFC** is average fixed cost. AFC(q) = FC/q. AFC(0) is
infinite and AFC(inf.) is zero. (yes, I should have said that the
limit as q approaches infinity of AFC is zero, but I am sure you
can tolerate the short hand.)

**MC(q)** is marginal cost. It is the cost of making the
next unit. MC(q) is approximately C(q+1) - C(q). Put the other
way, C(q+1) is approximately C(q) + MC(q). The cost of making q+1
units is the cost of making q units plus marginal cost at q. A Diagram.

Assuming that FC are not zero, AC(0) will be infinite and AC(inf) = AVC(inf)

Whenever AC is increasing, MC is above AC. When decreasing MC is below AC.

AC(q+1) - AC(q) = c(q+1)/(q+1) - c(q)/q = c(q)/(q+1) + mc(q)/(q+1) - c(q)/q

= -c(q)/(q(q+1) + mc(q)/q+1)

= (1/(q+1)) (mc(q) - ac(q) ) so mc greater than ac means ac(q+1) > ac(q) or ac increasing.

This means that **MC goes through the minimum point of AC**.
Note that MC = VC(q+1) - VC(q). (why?) so the above proof also
shows that **MC goes through the minimum point of AVC.**

Firms maximize profits. p = p q - c(q) = q (p - ac(q) ).

A necessary condition for maximizing profits is that **p =
mc(q)**. We assume mc slopes up.

Part of a proof: We will show that if p = mc(q*) then profits go down if one decreases q* by one unit. One can show, but we will not, that increasing q above q* will also decrease profits. Thus q* is a local profit maximum. Our claim is that

p(q*) - p(q*-1) is positive. Now p(q*) - p(q*-1) =

[pq* - c(q*)] - [ p (q*-1) - c(q*-1)]

= p - [ c(q*) - c(q*-1) ]

= p - mc(q*-1)

which is positive because mc slopes up and equals p at q*. (draw the picture)

(at home construct the same argument for for q*+1 and q*+2. do you see the problem with the discrete approximation of mc as c(q*+1) - c(q*) rather than the calculus answer of mc is limt -> 0 [c(q+t) - c(q)]/t.)

The **shutdown point**** **is
given by p = minimum point on avc. When p = min avc = mc ,
profits will be exactly - FC. p = q (
p - ac) = q ( p - avc) - fc but p = avc so p
= -fc. Profits would be less if p were less so p less than min
avc = mc, best policy is shutdown.

The **supply curve** is mc above avc.